We discuss the use of atom interferometry as a tool to search for dark matter (DM) composed of virialized ultralight fields (VULFs). Previous work on VULF DM detection using accelerometers has considered the possibility of equivalence-principle-violating effects whereby gradients in the dark matter field can directly produce relative accelerations between media of differing composition. In atom interferometers, we find that time-varying phase signals induced by coherent oscillations of DM fields can also arise due to changes in the atom rest mass that can occur between light pulses throughout the interferometer sequence as well as changes in Earth's gravitational field. We estimate that several orders of magnitude of unexplored phase space for VULF DM couplings can be probed due to these new effects.